Band-gap voltage reference circuits provide an output voltage that remains substantially constant over a wide temperature range. These reference circuits operate using the principle of adding a first voltage with a positive temperature coefficient to a second voltage with an equal but opposite negative temperature coefficient. The positive temperature coefficient voltage is extracted from a bipolar transistor in the form of the thermal voltage, kT/q (V.sub.T), where k is Boltzman's constant, T is absolute temperature in degrees Kelvin, and q is the charge of an electron. The negative temperature coefficient voltage is extracted from the base-emitter voltage (V.sub.BE) of a forward-biased bipolar transistor. The band-gap voltage, which is insensitive to changes in temperature, is realized by adding the positive and negative temperature coefficient voltages in proper proportions.
A conventional prior art band-gap circuit is shown in FIG. 1. In prior art circuits such as this, all the resistors are manufactured similarly, so the ratio of R3 20 to R4 30 would remain constant with respect to temperature. An operational amplifier 10 maintains an equal voltage across R3 20 and R4 30, thereby keeping the ratios of currents (IC1 to IC2) into the collectors of Q1 40 and Q2 50 equal over temperature also. It can be seen that IC1 is inversely proportional to R3 and current IC2 is inversely proportional to R4 30. The emitter areas of transistors Q1 40 and Q2 50 are in a ratio of A to nA with the emitter area of Q2 50 scaled larger than that of Q1 40 by a factor of n. The resulting collector currents and base to emitter voltages of the two transistors result in a voltage across R1 that equals kT/q ln(n×IC1/IC2), where ln is the natural logarithm function and n is the factor by which the emitter area of Q2 50 is scaled larger than that of Q1 40. The voltage across R1 is amplified across R2 by the factor of 2×R2/R1.
The band-gap circuit functions by taking output voltages that are positively and negatively changing with respect to temperature, and adding them to obtain a substantially constant output voltage with respect to temperature. Specifically, the base to emitter voltage, V.sub.BE of Q1 40 has a negative temperature coefficient, while the voltage across R2 has a positive temperature coefficient. By taking the output voltage of the circuit at the base of Q1 40, the positive and negative temperature coefficients essentially cancel, so the output voltage remains constant with respect to temperature.
A first-order analysis of a band-gap reference circuit approximates the positive and negative temperature coefficient voltages to be exact linear functions of temperature. The positive temperature coefficient voltage generated from V.sub.T is in fact substantially linear with respect to temperature. The generated negative temperature coefficient voltage from the V.sub.BE of a bipolar transistor contains higher order non-linear terms that have been found to be approximated by the function Tln(T), where ln(T) is the natural logarithm function of absolute temperature. When the band-gap voltage is generated using conventional circuit techniques, the Tln(T) term remains and is considered an error term which compromises the accuracy of the reference output voltage.
What is needed is a more accurate band-gap reference circuit that corrects for errors resulting from temperature changes that lead to errors in the reference voltage.